scholarly journals A step beyond Freiman’s theorem for set addition modulo a prime

2020 ◽  
Vol 32 (1) ◽  
pp. 275-289
Author(s):  
Pablo Candela ◽  
Oriol Serra ◽  
Christoph Spiegel
Keyword(s):  
Set Addition ◽  
Freiman’S Theorem

scholarly journals Restricted set addition in groups, II. A generalization of the Erdős-Heilbronn conjecture

10.37236/1482 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Vsevolod F. Lev
Keyword(s):  
General Type ◽  
Sharp Estimate ◽  
Abelian Groups ◽  
Polynomial Method ◽  
Set Addition

In 1980, Erdős and Heilbronn posed the problem of estimating (from below) the number of sums $a+b$ where $a\in A$ and $b\in B$ range over given sets $A,B\subseteq{\Bbb Z}/p{\Bbb Z}$ of residues modulo a prime $p$, so that $a\neq b$. A solution was given in 1994 by Dias da Silva and Hamidoune. In 1995, Alon, Nathanson and Ruzsa developed a polynomial method that allows one to handle restrictions of the type $f(a,b)\neq 0$, where $f$ is a polynomial in two variables over ${\Bbb Z}/p{\Bbb Z}$. In this paper we consider restricting conditions of general type and investigate groups, distinct from ${\Bbb Z}/p{\Bbb Z}$. In particular, for $A,B\subseteq{\Bbb Z}/p{\Bbb Z}$ and ${\cal R}\subseteq A\times B$ of given cardinalities we give a sharp estimate for the number of distinct sums $a+b$ with $(a,b)\notin\ {\cal R}$, and we obtain a partial generalization of this estimate for arbitrary Abelian groups.

scholarly journals Large values of the additive energy in and

2014 ◽  
Vol 156 (2) ◽  
pp. 327-341 ◽  
Author(s):  
XUANCHENG SHAO
Keyword(s):  
Arithmetic Progression ◽  
Upper Bounds ◽  
Large Piece ◽  
Small Rank ◽  
Optimal Bound ◽  
Additive Energy ◽  
Freiman’S Theorem ◽  
Proof Strategy ◽  
Generalized Arithmetic

AbstractCombining Freiman's theorem with Balog–Szemerédi–Gowers theorem one can show that if an additive set has large additive energy, then a large piece of the set is contained in a generalized arithmetic progression of small rank and size. In this paper, we prove the above statement with the optimal bound for the rank of the progression. The proof strategy involves studying upper bounds for additive energy of subsets of ${\mathbb{R}^d$ and ${\mathbb{Z}^d$.

scholarly journals Equality in Pollard's theorem on set addition of congruence classes

2007 ◽  
Vol 127 (1) ◽  
pp. 1-15 ◽  
Author(s):  
E. Nazarewicz ◽  
M. O'Brien ◽  
M. O'Neill ◽  
C. Staples
Keyword(s):  
Set Addition ◽  
Congruence Classes

scholarly journals Freiman's Theorem in Finite Fields via Extremal Set Theory

2009 ◽  
Vol 18 (3) ◽  
pp. 335-355 ◽  
Author(s):  
BEN GREEN ◽  
TERENCE TAO
Keyword(s):  
Set Theory ◽  
Finite Fields ◽  
Common Theme ◽  
Additive Combinatorics ◽  
Extremal Set Theory ◽  
Extremal Set ◽  
Freiman’S Theorem ◽  
Sharp Version

Using various results from extremal set theory (interpreted in the language of additive combinatorics), we prove an asymptotically sharp version of Freiman's theorem in $\F_2^n$: if $A \subseteq \F_2^n$ is a set for which |A + A| ≤ K|A| then A is contained in a subspace of size $2^{2K + O(\sqrt{K}\log K)}|A|$; except for the $O(\sqrt{K} \log K)$ error, this is best possible. If in addition we assume that A is a downset, then we can also cover A by O(K46) translates of a coordinate subspace of size at most |A|, thereby verifying the so-called polynomial Freiman–Ruzsa conjecture in this case. A common theme in the arguments is the use of compression techniques. These have long been familiar in extremal set theory, but have been used only rarely in the additive combinatorics literature.

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